Introduction:
To find the smallest number by which 7436 must be multiplied to get a perfect square, we need to analyze its prime factorization. By decomposing 7436 into its prime factors and examining their powers, we can determine the smallest number that will result in a perfect square.

Prime Factorization of 7436:
To find the prime factorization of 7436, we start by dividing it by the smallest prime number, which is 2. The result is 3718, which is an even number. We continue dividing by 2 until we obtain an odd number. The prime factorization of 7436 is as follows:

7436 = 2 × 2 × 1859

Finding the Smallest Number for Perfect Square:
To make a perfect square, the exponents of each prime factor in the factorization should be even. Therefore, we need to determine the smallest number that will result in an even exponent for each prime factor.

Prime Factorization Analysis:
Let's analyze the prime factorization of 7436:

- The prime factor 2 appears twice in the factorization, which means its exponent is 2.
- The prime factor 1859 appears once in the factorization, which means its exponent is 1.

Calculating the Smallest Number:
To make the exponent of each prime factor even, we need to multiply 7436 by the smallest number that will raise the exponent to an even value.

- For the prime factor 2, we already have an exponent of 2, which is even.
- For the prime factor 1859, we need to multiply 7436 by 1859 to make its exponent 2.

Therefore, the smallest number by which 7436 must be multiplied to get a perfect square is 7436 × 1859 = 13,801,724.

Conclusion:
To find the smallest number by which 7436 must be multiplied to get a perfect square, we analyzed its prime factorization. By ensuring that the exponent of each prime factor is even, we determined that the smallest number is 13,801,724.

7436 is multiplied by 11 then it is a perfect square